In his first inaugural address in 2014, Mayor Bill de Blasio invoked the names of legendary reformers, from Fiorello La Guardia to Franklin and Eleanor Roosevelt, and said that in their honor and spirit, he would “commit to a new progressive direction in New York.”

The anticipation for many of Mr. de Blasio’s supporters was particularly keen because it built on the sense that his predecessor, Michael R. Bloomberg, had presided over the city’s growth in a way that benefited the well-off and left the poor behind.

But midway through his fifth year in office, Mr. de Blasio has disappointed some of his most loyal backers, who point to a range of issues, from criminal justice reform and homelessness to the protection of immigrants, where the mayor has fallen short of his promises.

On Monday, Mr. de Blasio found himself having to explain why, after enacting a policy to end solitary confinement in city jails, the city Department of Correction increased the number of inmates shipped upstate, where they were put in solitary. He defended the actions as rare and necessary for safety.

Other promises have also run up against reality.

The city has not succeeded in opening new neighborhood-based homeless shelters as fast as Mr. de Blasio had vowed.

His recurring pledge to raise taxes on high-earners has been dashed in Albany, where the mayor holds little sway and has been embroiled in a yearslong feud with Gov. Andrew M. Cuomo.

And Mr. de Blasio has been resolute in his opposition to congestion pricing as a way to finance much-needed mass transit improvements, frustrating many progressive advocates who see his position as inconsistent with his goal of narrowing inequity.

“I think he’s run into the mechanics of governing, where you have to deal with the restraints of law and City Council and dealing with the practicalities of things like union negotiations,” the Rev. Al Sharpton said in an interview. “I don’t think he’s had a problem of commitment. I think he has a problem of having to govern, restraints that sometimes are troubling to those that are nongovernmental activists.”

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Mr. de Blasio spoke at a town hall at a school in Bedford-Stuyvesant, Brooklyn, in March.CreditSam Hodgson for The New York Times

Offered a list on Thursday of frequent criticisms from progressive advocates of his record, including expanding the secrecy surrounding certain Police Department records and failures at the New York City Housing Authority, Mr. de Blasio said: “That’s kind of the world upside down.”

“I’ve been a progressive my whole life, know a lot about the left,” he said. “I’ve talked to a whole lot of progressives about what we’re doing, activists, elected officials, issue experts. And I have found a lot of agreement with our agenda.”

The mayor is banking on it: He announced the creation of a political action committee, Fairness PAC, in order to fund national Democratic candidates, and also pay for political travel outside of New York City by Mr. de Blasio and his wife, Chirlane McCray. His former campaign website was relaunched to push the effort, which was reported on Thursday by Politico.

Mr. de Blasio defended his record as a progressive leader, saying that there will always be advocates “who want us to take additional steps, and we intend to.” He said that on balance, his efforts to improve police discipline, end solitary confinement for young inmates and create neighborhood homeless shelters were “progressive and tangible” actions that affect “thousands and thousands” of people’s lives.

“There may be a few advocates who see it differently,” the mayor said. “But the vast majority of New Yorkers if you ask them, ‘Hey, this guy ended solitary confinement for these many thousands of inmates versus all the people before who tolerated it.’ What’s more progressive?”

To be sure, the mayor has made good on many of his commitments, including pursuing an aggressive affordable housing plan, launching a city-subsidized ferry service and the expansion of early childhood education.

Mr. de Blasio’s greatest accomplishment has been the creation of a new universal prekindergarten program, building on that success by expanding it to 3-year-olds. Yet progressives have faulted him for not tackling the deep-seated problem of segregation in the city’s school system.

His push to build new affordable units or preserve those that were at risk of shifting to market-rate rents has been met with criticism from housing advocates who say not enough of the apartments are affordable for the poorest New Yorkers.

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Last week, the mayor visited Cayuga Centers in East Harlem, where some of the migrant children taken from their parents at the border were being housed in New York City. About two-thirds of the children who had been in New York left the city on Wednesday and Thursday to be reunited with parents.CreditHiroko Masuike/The New York Times

Mr. de Blasio also has found himself repeatedly outflanked on the left by upstart political figures like Alexandria Ocasio-Cortez, the Democratic socialist who upset Representative Joseph Crowley last month in the New York Democratic congressional primary, and the actress Cynthia Nixon, a Democratic candidate for governor.

Ms. Nixon, for example, backs marijuana legalization; Mr. de Blasio refuses to go that far, although he has taken steps to end the practice of arresting most people caught smoking marijuana in public.

The City Council, too, has pressured the mayor from the left. In his first term, Melissa Mark-Viverito, then the Council speaker, pushed him to endorse the eventual closing of Rikers Island; in his second term, the new speaker, Corey Johnson, forced him to fund discounted subway and bus rides for poor New Yorkers.

Similarly, his effort during the 2016 campaign to spearhead a national movement around left-of-center policies faltered. He has disappointed many in the left flank of the Democratic Party with his cautious endorsements in pitched Democratic primary fights: Hillary Clinton over Bernie Sanders in the 2016 presidential race; Mr. Cuomo over Zephyr Teachout in the 2014 Democratic primary for governor; Mr. Crowley, against Ms. Ocasio-Cortez, earlier this year.

On immigration, advocates saw the mayor, who has championed New York as a sanctuary city in the age of President Trump, as an ally for much of his first term: He oversaw the creation of a city I.D. card that could be used by immigrants regardless of their legal status and he supported a spending increase for literacy and adult education classes.

But last year, as he ran for re-election, Mr. de Blasio changed a policy that advocates considered essential when he said that the city would no longer pay for legal representation for all people in immigration court facing deportation. For the first time, Mr. de Blasio refused to provide city money to pay for immigration lawyers for people who had been convicted of any of 170 crimes, including violent offenses and drug crimes.

A compromise was worked out where the City Council agreed to have private donors finance cases not covered by city money, but advocates were stung by what they saw as Mr. de Blasio’s reversal.

“It continues to be a stain in his legacy on immigration,” said Javier H. Valdés, a co-executive director of Make the Road New York, an advocacy group. “There’s no circle in the immigrant rights community where people don’t talk about that.”

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Mr. de Blasio has proposed a so-called millionaire’s tax to pay for improvements to the city’s beleaguered subway system.CreditSam Hodgson for The New York Times

On policing issues, those pushing Mr. de Blasio from the left have been more caustic in their evaluation, observing that even while low-level arrests are declining, the people being arrested remain mostly black and Hispanic. At the same time, advocates said, the mayor has not done enough to ensure bad-acting cops are punished.

She added that the mayor had “taken the city back decades on police transparency with newly expanded interpretations of 50-a that are worse than the Giuliani and Bloomberg administrations,” referring to a state law that protects certain police documents from disclosure. Mr. de Blasio has vowed to fight for a change to the 50-a law in Albany, but little lobbying on the issue has been evident.

Nick Sifuentes, the executive director of the Tri-State Transportation Campaign, an advocacy group of transportation and mass transit issues, said he initially saw Mr. de Blasio as a “progressive warrior” when he came into office, “saying all the right things about what New York really needed.” But the mayor’s first term has left him “very frustrated at the slow pace” of mayoral action on key transit initiatives, even as Mr. de Blasio has been an effective champion through his Vision Zero approach to trying to reduce traffic injuries and deaths.

Marcia Bystryn, the president of the New York League of Conservation Voters, too, said that the city had been a leader on environmental issues and praised Mr. de Blasio for setting ambitious goals.

Even so, one progressive goal — bringing its brown-bin composting program citywide by the end of the year — will fall short. “We paused the program,” the sanitation commissioner, Kathryn Garcia, said in an interview, adding that no new deadline had been set. “I know that the mayor remains very committed to getting this done, but I think that he was trying make sure that he was balancing the cost concerns.”

Ms. Bystryn suggested that the mayor “take a good hard look at his very ambitious OneNYC plan and focus on the practical steps that will be necessary to implement those goals. And if there are no practical steps, then adjust the goals.”

Mr. Sharpton said that some of the frustration with Mr. de Blasio among his far-left base may stem from unrealistic expectations.

“A lot of people expected him to be more of what they projected on him than what he was,” Mr. Sharpton said. “I knew when he was coming in that he was a deliberate, very careful type of guy.”

A version of this article appears in print on , on Page A1 of the New York edition with the headline: Progressives Who Once Hailed De Blasio Bemoan Slow Gains. Order Reprints | Today’s Paper | Subscribe